Total Marks – 200
Paper I
Marks – 100

Candidates will be asked to attempt three questions from Section A and two questions from Section B
Section A
Modern Algebra
1. Groups, subgroups, languages, theorem, cyclic groups, normal sub-groups, quotient groups,
fundamental theorem of homomorphism. Isomorphism theorems of groups, inner
automorphosms. Conjugate elements, conjugate sub-groups, commutator sub-groups
2. Rings, sub rings, integral domains, quotient fields, isomorphism theorems, field extension and
finite fields
3. Vector spaces, linear independence, bases, dimensions of a finitely generated space, linear
transformations, matrices and their algebra. Reduction of matrices to their echelon form. Rank
and nullity of a linear transformation
4. Solution of a system of homogenous and non-homogenous linear equations. Properties of
determinants. Cayley-Hamilton theorem, Eigen values and eigenvectors. Reduction to canonical
forms, specially digitalization
Section B
1. Conic sections in Cartesian coordinates, Plane polar coordinates and their use to represent the
straight line and conic sections. Cartesian and spherical polar coordinates in three dimension.
The plane, the sphere, the ellipsoid, the paraboloid and the hyperboloid in Cartesian and
spherical polar coordinates
2. Vector equations for plane and for space-curves. The arc length. The osculating plane. The
tangent, normal and bi-normal. Curvature and torsion. Serre-Frenet’s formulae. Vector equations
for surfaces. The first and second fundamental forms. Normal, principal, Gaussian and mean
Paper II
Marks – 100

Candidates will be asked to attempt any three questions from Section A and two questions from Section
Section A
Calculus and Real Analysis
1. Real numbers, limits, continuity, differentiability, indefinite integration, mean value theorems.
Taylor’s theorems, indeterminate form. Asymptotes, curve tracing, definite integrals, functions of
several variables. Partial derivates. Maxima and minima. Jacobeans, double and triple integration
(Techniques only). Applications of Beta and Gamma functions. Areas and volumes. Riemann-
Stieltje’s integral. Improper integrals and their conditions of existence. Implicit function theorem.
Absolute and conditional convergence of series of real terms. Rearrangement of series, uniform
convergence of series
2. Metric spaces. Open and closed spheres. Closure, interior and exterior of a set
3. Sequence in metric space. Cauchy sequence, convergence of sequences, examples, complete
metric spaces, continuity in metric spaces. Properties of continuous functions
Section B
Complex analysis
Function of a complex variable, Demoiver’s theorem and its applications. Analytic functions, Cauchy’s
theorem. Cauchy’s integral formula, Taylor’s and Laurent’s series. Singularities. Cauchy residue theorem
and contour integration. Fourier series and Fourier transforms. Analytic continuation


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